So, I'm having a hard time wrapping my mind around the storm rules.

It says in the "fluff" that the storm quality doubles the number of hits inflicted on the target, and the ammount of ammo expended. The part about each degree of sucees giving two hits instead of one, I understand. But when you compare a storm bolter to a regular godwyn pattern bolter, they have the same RoF of S/2/4. Obviously this means that the storm bolter has a capacity to do 8 hits as opposed to 4 from the normal one. Based on the description though, it suggests that a storm bolter burns through ammo as if it had the RoF of 2/4/8. Is this correct? Or does it still go through ammo at a rate of 1/2/4, and those double hits somehow magically happen because of two barrels?

You are correct that the RoF of S/2/4 is equivalent to 2/4/8 for the purposes of ammo consumption.